Page 48 - Schaum's Outline of Theory and Problems of Advanced Calculus

Functions, Limits, and

Continuity

FUNCTIONS
A function is composed of a domain set, a range set, and a rule of correspondence that assigns
exactly one element of the range to each element of the domain.
This definition of a function places no restrictions on the nature of the elements of the two sets.
However, in our early exploration of the calculus, these elements will be real numbers. The rule of
correspondence can take various forms, but in advanced calculus it most often is an equation or a set of
equations.
If the elements of the domain and range are represented by x and y, respectively, and f symbolizes
the function, then the rule of correspondence takes the form y ¼ f ðxÞ.
The distinction between f and f ðxÞ should be kept in mind. f denotes the function as defined in the
first paragraph. y and f ðxÞ are different symbols for the range (or image) values corresponding to
domain values x. However a ‘‘common practice’’ that provides an expediency in presentation is to read
f ðxÞ as, ‘‘the image of x with respect to the function f ’’ and then use it when referring to the function.
(For example, it is simpler to write sin x than ‘‘the sine function, the image value of which is sin x.’’)
This deviation from precise notation will appear in the text because of its value in exhibiting the ideas.
The domain variable x is called the independent variable. The variable y representing the corre-
sponding set of values in the range, is the dependent variable.
Note: There is nothing exclusive about the use of x, y, and f to represent domain, range, and
function. Many other letters will be employed.
There are many ways to relate the elements of two sets. [Not all of them correspond a unique range
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value to a given domain value.] For example, given the equation y ¼ x, there are two choices of y for
each positive value of x.As another example, the pairs ða; bÞ, ða; cÞ, ða; dÞ, and ða; eÞ can be formed and
again the correspondence to a domain value is not unique. Because of such possibilities, some texts,
especially older ones, distinguish between multiple-valued and single-valued functions. This viewpoint
is not consistent with our definition or modern presentations. In order that there be no ambiguity, the
calculus and its applications require a single image associated with each domain value. A multiple-
valued rule of correspondence gives rise to a collection of functions (i.e., single-valued). Thus, the rule
2
y ¼ x is replaced by the pair of rules y ¼ x 1=2 and y ¼ x 1=2 and the functions they generate through the
establishment of domains. (See the following section on graphs for pictorial illustrations.)
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